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[For the entire collection see Zbl 0657.00005.] \\par Let X(N) be a modular curve of level N\\in \\bbfN; \\frak x be a divisor of degree 0 on X(N); J\\sb X be the Jacobian of X; J\\sb X(\\bbfQ) be the set of its rational points; lt;.,.gt; be the height pairing on J\\sb X(\\bbfQ) and h(\\frak x)=lt;\\frak x,\\frak xgt; be the height of \\frak x. For a cusp form f(z)=\\sum\\sp∞\\sbn=1a\\sb f(n)⋅ \\exp (2π inz) on X(N), the L-series L(f,s):=\\sum\\sp∞\\sbn=1a\\sb f(n)⋅ n\\sp-s is defined and it is a meromorphic function in s. The problem is the following: construct on the modular curve X(N) explicit divisors of degree 0 defined over \\bbfQ, and relate their heights to the derivatives at s=1 of L-series of cusp forms of weight 2 and level N. The author obtains the following theorem: \\par Let D be the discriminant of an imaginary quadratic field and let \\frak x\\sb D be the divisor defined for D. Assume N is prime. Then h(\\frak x\\sb D)=\\vert D\\vert\\sp-1/2(4π\\sp 2)\\sp-1\\sum\\sbf\\Vert f\\Vert\\sp-2L'(f,1)L(f,D,1) where the sum runs over all Hecke forms with w\\sb f=-1 and L(f,D,1) is the value at s=1 of the ``twisted L- series'' L(f,D,s)=\\sum\\sp∞\\sbn=1a\\sb f(n)⋅ (\\fracDn)n\\sp-s. \\par The author also obtains an expression of lt;\\frak x\\sb D,\\frak x\\sbD'gt; for two divisors \\frak x\\sb D and \\frak x\\sb D in terms of the values of the L-series. The Néron-Tate height is a sum of local contributions from all places of \\bbfQ and it is necessary to compute all local heights in the proof of the above theorems. \\par In the second part of this paper some of the arithmetic questions which arise in this context are discussed.
